nLab
picture of mechanics

Idea

Dynamics affects both observables and, dually, states; this is most well known in quantum mechanics but applies equally well to classical mechanics. The different “pictures” of mechanics differ in how the dynamics is explicitly formalized:

• In the Heisenberg picture, the dependence of observables on time? (or more generally spacetime) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the Haag–Kastler axioms of AQFT.

• In the Schrödinger picture, states are propagated through time?, while observables are held fixed; the axiomatic formalization of this is given by cobordism category representations in FQFT.

• The Dirac (interaction) picture is a mixture of these two approaches: dynamics is split into a free? (or otherwise solvable?) part and an interaction or perturbation; one of these is take to affect the states, the other the observables.

The pictures are named after those physicists (Werner Heisenberg, Erwin Schrödinger, and Paul Dirac) who first used or popularised these approaches to quantum physics.

Formulation

With global time

Let us assume a global notion of time?, say a fixed background? spacetime which is globally hyperbolic, so that it admits a foliation into Cauchy surfaces, and choose a time coordinate? for this foliation. The upshot of this is that each event? occurs at a time $t$, and conversely we can speak of space at any time $t$ (at least within certain bounds). Thus we may speak sensibly of either the state of the world at time $t$ or the value of some observable quantity at time $t$.

Because this is a picture of dynamics, states or observables (as appropriate to the picture) will vary through time. We therefore have a time evolution? operator $U(t,t\prime )$ between any two times $t,t\prime$; actually, we need consider only $U(t)≔U(t,0)$, since $U(t,t\prime )=U(t)\circ U(t\prime {)}^{-1}$.

In the Heisenberg picture, for each observable $A$, we speak of $A$ only at some time $t$, so our actual observables are of the form $A(t)$. We write abstractly

$$A(t)=A(0)\cdot U(t)$$A(t) = A(0) \cdot U(t)

to show the evolution of the observable through time. However, when it comes to the state of the world, we speak of a single state $\psi$ that describes the world at all times.

In the Schrödinger picture, we instead speak of $\psi (t)$, the state of the world at time $t$. We write abstractly

$$\psi (t)=U(t)*\psi (0)$$\psi(t) = U(t) \ast \psi(0)

(the Schrödinger equation) to show the evolution of the state through time. However, when it comes to observables, we use only the observable $A$ across all times.

To see the connection between the two pictures, recall that an observable $A$ and a state $\psi$ together produce a probability distribution giving the probability that any given value of $A$ will be observed, given that the world is in state $\psi$. (This is true throughout mechanics, although it is obscured in non-statistical classical mechanics, since the probability distributions produced by classical pure states are all delta measure?s.) Assuming that $A$ belongs to an appropriate algebra of observables and the probability measures are sufficiently nice, we may restrict attention to the expectation values $⟨A{⟩}_{\psi }$ of these distributions, since the entire distribution can be recovered from $⟨A^n{⟩}_{\psi }$ as $n$ varies over natural numbers.

The connection between the two pictures is then given by

$$⟨A\cdot U(t){⟩}_{\psi }=⟨A{⟩}_{U(t)*\psi }.$$\langle{A \cdot U(t)}\rangle_\psi = \langle{A}\rangle_{U(t) \ast \psi} .

It remains to say exactly what $U(t)$ is and what the operations $\cdot$ and $*$ are. Let us use the density matrix formulation of quantum statistical mechanics?, since classical and non-statistical mechanics may be recovered as special cases, by restricting (respectively) the allowed observables or states. In this case, both states and observables are given by linear operators on a Hilbert space $H$, and we have

$$⟨A{⟩}_{\psi }=tr(A\psi )$$\langle{A}\rangle_\psi = \tr(A \psi)

(using the trace operation). Each $U(t)$ is a unitary operator on $H$ (since time evolution between Cauchy surfaces is a symmetry), and we have

$$A\cdot U(t)=U(t{)}^{-1}AU(t)$$A \cdot U(t) = U(t)^{-1} A U(t)

(a right action?) and

$$U(t)*\psi =U(t)\psi U(t{)}^{-1}$$U(t) \ast \psi = U(t) \psi U(t)^{-1}

(a left action?). We then have

$$⟨A\cdot U(t){⟩}_{\psi }=\mathrm{tr}(U(t{)}^{-1}AU(t)\psi )=\mathrm{tr}(AU(t)\psi U(t{)}^{-1})=⟨A{⟩}_{U(t)*\psi },$$\langle{A \cdot U(t)}\rangle_\psi = tr(U(t)^{-1} A U(t) \psi) = tr(A U(t) \psi U(t)^{-1}) = \langle{A}\rangle_{U(t) \ast \psi} ,

as desired, using the cyclic property of the trace?.

The time evolution operator $U(t)$ is often derived from a Hamiltonian and the formula for $A(t)$ or $\psi (t)$ is further derived from a differential equation involving this Hamiltonian. However, this is unnecessary for the connection between the two pictures.

Without time

If spacetime is not globally hyperbolic, then there is no time coordinate $t$, and none of the discussion above makes sense; or if we choose a coordinate $t$ and call it time regardless, then time evolution is not a symmetry and we do not have the operators $U(t)$.

In this case, the Heisenberg picture still makes sense, even though we cannot expect to calculate $A(t)$ from $A(0)$ (if it even makes sense to discuss such things). This is easily seen in field theory, where the operators called $A$ above are really of the form $A(x,y,z)$. Then the Heisenberg picture's $A(t)$ is really $A(x,y,z,t)$, or simply $A(p)$ where $p$ indicates an event? (a point in spacetime). So even if the coordinates $x,y,z,t$ do not make sense, still $A(p)$ does; and even if the equations of physics cannot be thought of as describing evolution through time, still they can be thought of as describing the relationships between observables at different places in spacetime.

In contrast, the Schrödinger picture cannot be so treated. One may be led to the contrary impression by the quantum mechanics of a single particle without any internal structure (not even spin), in which case the Hilbert space of (pure quantum-mechanical) states is naturally identified with $L^2({ℝ}^3)$ and the state $\psi$ is really $\psi (x,y,z)$. In this case, the Schrödinger picture's $\psi (t)$ is really $\psi (x,y,z,t)$, that is $\psi (p)$. However, this fails in classical or statistical mechanics; and even in non-statistical quantum mechanics, it breaks down if the particle has internal structure or there is more than one particle in the world. Then we see that the spacial coordinates $x,y,z$ generalise to the arbitrary coordinates of configuration space, while $t$ remains only $t$, and there is no way to subsume it into a spacetime coordinate.

History

Historically, the terms ‘Schrödinger picture’ and ‘Heisenberg picture’ (at least) referred to more than what we discuss above; they referred to the entirety of the differences between Schrödinger's and Heisenberg's approaches to quantum mechanics.

For example, these terms included also Schrödinger's use of typically wave-like functions as pure states (and correspondingly operators in the higher-type-theoretic sense as observables) vs Heisenberg's use of infinite-dimensional matrices as observables (and correspondingly infinite sequences as pure states). This difference was rectified by von Neumann's application of Hilbert space to the problem, showing that (if one suitably restricts the allowed functions and sequences and also identifies equivalent functions a bit) both approaches used Hilbert space (what we would now call the infinite-dimensional separable Hilbert space) as the space of pure states.

This is entirely separate from the question of whether states or observables are taken to evolve with time. Still, there is this connection: Schrödinger evolved states, and his approach was called ‘wave mechanics’ after his representation for states, while Heisenberg evolved observables, and his approach was called ‘matrix mechanics’ after his representation for observables.

Related concepts

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

References

See for instance sections 7.19.1–3 in

• Eberhard Zeidler, Quantum field theory. A bridge between mathematicians and physicists – volume I Springer (2009) (web).

To check conventions at least, see Wikipedia:

• Heisenberg picture,
• Schrödinger picture.

A note on how the Schrödinger picture in the form of extended FQFT on Lorentzian manifolds is related to the Heisenberg picture in the form of AQFT is in

• Urs Schreiber, AQFT from n-Functorial QFT , Communications in Mathematical Physics, Volume 291, Issue 2 (2009), pp.357-401